A new semi-analytical technique for nonlinear systems based on response sensitivity analysis

Abstract

In this paper, a new semi-analytical method, namely the time-domain minimum residual method, is proposed for the nonlinear problems. Unlike the existing approximate analytical method, this method does not depend on the small parameter and can converge to the exact analytical solutions quickly. The method is mainly threefold. Firstly, the approximate analytical solution of the nonlinear system ${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$ is expanded as the appropriate basis function and a set of unknown parameters, i.e., ${\varvec{x(t)}}\approx \sum _{i=0}^{N}{\varvec{a_i\chi _i(t)}}$. Then, the problem of solving analytical solutions is transformed into finding a set of parameters so that the residual ${\varvec{R}}={\varvec{F}}\left( \sum _{i=0}^{N}a_i\ddot{\chi }_i,\sum _{i=0}^{N}a_i{\dot{\chi }}_i,\sum _{i=0}^{N}a_i\chi _i\right)$ is minimum over a period, i.e., $\underset{{\varvec{a}}\in {\mathscr {A}}}{\min }\int _{0}^T {\varvec{R}}({\varvec{a}},t)^{T} {\varvec{R}}({\varvec{a}},t) \mathrm {d} t$. The nonlinear equation ${\varvec{F\left( \ddot{x},{\dot{x}},x\right) }}={\varvec{0}}$ is regarded as the objective function to optimize, and the process of solving the analytic solution is transformed into a nonlinear optimization process. Finally, the optimization process is iteratively solved by the enhanced response sensitivity approach. Four numerical examples are employed to verify the feasibility and effectiveness of the proposed method.